Nonlinear Problems in Elasticity
Course description
The course is an advanced introduction to nonlinear problems of elastic continuum media. It will start with a review of basic principles and equations for linear elasticity in continuum media under the small strains assumptions. It will then introduce slender structures, i.e. rods and plates and analyze buckling and instabilities introduced by the geometric nonlinearity of their kinematics. Bringing kinematic nonlinearities to three dimensional to continuum structures will naturally lead to finite strains and nonlinear elasticity. The theoretical topics will be illustrated by workout examples and experiments using rods, strings, rubber balloons and sheets. The concepts are introduced using precise mathematical deductions and physical assumptions and will be discussed in depth. However, calculations will be kept to a minimum by using symbolic and numeric computations to solve different problem settings. The presented method can be applied for various engineering applications, e.g. the prediction of the failure of structures by buckling and instability, the design of novel materials with architectured microstructures or the analysis of soft biological materials.Time and place
Tuesday, 10:15  13:00 (Lecture&Exercise),LEE C 104Lecturer
Prof. Dr. Andrei ConstantinescuOpendoor policy or by appointment, CLA J 27
Teaching assistant
Jinwoong ChaOpendoor policy or by appointment, CLA J 15.1
Syllabus
Date  Lecture  Exercise 
23.02. 
Lecture #1: Motivation, Euler Elastica Calculus of variations Slide Note Mathematica notebook 
Exercise #1: Calculus of variations Exercise #1 Further reading 
01.03. 
Lecture #2:Linear Elasticity: basic equations and exact solutions Slide Note Mathematica notebook 
Exercise #2: Mechanics of curly hair Exercise #2 
08.03. 
Lecture #3:Structires in Equilibrium Note 

15.03. 
Lecture #4:Reciprocity or further variaonal calculus & Energy in Linear Elascity Slide 
Exercise #4 
22.03. 
Lecture #5:Finite strains, Nonlinear elasticity, Inflating a rubber ballon Slide 

05.04. 
Lecture #6:Review_Stress_LargeStrain, Prestrain_Torsion Note_large strain Note_torsion Paper 

12.04. 
Lecture #7:Rubber ballons and Stability Slide Note in French Paper 

19.04. 
Lecture #8:Rubber ballons and Stability 2 Slide 

26.04.  Lecture #9:Physical mechanism for material elasticity  
03.05.  Lecture #10:Nonlinear elasticity: basic equations and exact solutions  
10.05.  Lecture #11:Stretch of a rubber sheets  
17.05.  Lecture #12:Torsional instability  
24.05.  Lecture #13:Stability of many rubber balloons  
31.05.  Lecture #14: From instabilities to phase change: shape memory alloys 
Literature:
 Basile Audoly and Yves Pomeau Elasticity and geometry: from hair curls to the nonlinear response of shells, Oxford University Press, 2010
 Patrick Ballard and Alain Millard Poutres et arcs elastiques, Editions Ecole Polytechnique, 2009
 Andrei Constantinescu and Alexander Korsunsky Elasticity with Mathematica : An Introduction to Continuum Mechanics and Linear Elasticity , Cambridge University Press, 2007
 Yibin B Fu and Raymond W OgdenNonlinear elasticity: theory and applications, Cambridge University Press, 2001
 Gerhard A HolzapfelNonlinear solid mechanics, Wiley, 2000
 I Mueller and P StrehlowRubber and Rubber Balloons (Lecture Notes in Physics 637),Springer: Heidelberg,Germany, 2004
Performance assessment
Homework: 30%
Project work: 30%
Final Exam: 40%